For those avoiding looking so as not to spoil their own fun, I'll point out that the hexagon-to-triangle dissection shown on that page doesn't use straight cuts. (One cut has a kink in it.)
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BlueNov 24 '13 at 21:15

@Blue It looks like there may be one of the apparent line segments which is really two segmenmts making an angle near 180 degrees. Is that what a "kink" is? If so it only increases the count of cuts by 1, and still keeps the number of regions as 5.
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coffeemathNov 25 '13 at 0:19

@coffeemath: The kink I see in the "6-3" figure is a dramatic bend, making a clearly concave piece that wraps around one corner of the (apparently-equilateral) triangular piece in the hexagon arrangement.
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BlueNov 25 '13 at 0:28

The lines look straight to me although some start and stop at unspecified positions. It's not at all clear how you actually make the cuts. I feel a really good answer would come with a set of instructions for using a ruler and compass. Can this be done?
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marshallNov 25 '13 at 17:14